The physicist Wolfgang Pauli (the “Pauli” of “Pauli exclusion” and “Pauli spin” was, so the story goes, fond of dismissing half-formed ideas and half-baked articles which he was reviewing for publication by saying, “That’s not right. It’s not even wrong.” It was the harshest criticism he could give.

I am currently teaching a math course intended for future teachers, and I recently began a class with that quote from Pauli. In the context of physics, “not even wrong” meant that a paper, idea, or theory did not lead to any predictions that could be tested (is not “falsifiable”, an essential criterion for a scientific idea). However, “not even wrong” applies outside physics and science to those ideas that are so poorly articulated, so barely-formed, that there’s no meaningful way to respond to them. Every teacher, I reminded my class, will all too often find herself talking to a complaining student appealing for more points on a homework or an exam, demanding “Why is this wrong?”

If the teacher is lucky, there is an answer to that question. “It actually happened in 1965, not 1978.”, for example, or “You said ‘rhombus’; you probably meant ‘rectangle’.” A specific error of fact or reasoning that can be identified, corrected, responded to. Usually, though, that’s not the case; instead, the answer on the page is a word salad, some vocabulary from the course strewn haphazardly on the page in the hopes of some partial credit or lucking into the right answer. If the class is a math class, it may not even be a sentence or recognizable as English. And what’s so frustrating for the teacher in this scene is, there’s nothing to point to! There’s no particular wrong word or error in reasoning. What the student has done is not even wrong.

A few years ago, I got fed up with this situation arising while I was grading. I decided that, while I was happy to write extensive comments to explain mistakes, I had no intention of spending that kind of energy on someone who hadn’t bothered to make any sense. I started writing “intrangible” on such passages, on the idea that marking them with actual English words gave them too much credit.

I have heard that at least one former student of mine has done some teaching of his own, and uses the word “intrangible” in his own lectures and grading. How about that.

More touching actually was when I received a visit from a different former student. She had struggled mightily in my class, and had not found much success at that time. However, subsequent to my class, she had taken an interest in more sophisticated mathematics, and had managed to teach herself quite a lot. After exchanging pleasantries, she asked me a question that she had been thinking about. The question was complex, and subtle, and I had to think for several minutes before I could answer it. Indeed I was proud of her! But as I was thinking, she got nervous, and asked tentatively, “Oh dear… was that intrangible?”

I assured her it was the most trangible thing anyone had said to me all day.

P.S.

Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law is also the name of an excellent book (ISBN: 9780465092756) by Peter Woit on string theory for a wide audience. If you want a friendly introduction in layman’s terms to string theory and why it is and isn’t important, I recommend it. If you don’t have the time or attention span for a book, this xkcd comic makes essentially the same point in a second or two.

P.P.S.

When a student says to me “Why is that wrong?” with that challenge in his voice, of course the temptation is to counter-challenge “Why would you think that it’s right?” (Or “Why would you think that’s a sentence?”, if I’m at my most exasperated with what some student think passes for acceptable writing skills in math class.) Sadly, a teacher is so often prevented by etiquette from saying what would be the most apt thing. (Another thing I wish I could say sometimes, when a student comes to my office demanding help with a homework problem, having admitted or demonstrated that he hasn’t really thought about it yet, or sometimes hasn’t even read the whole thing, is “Have you tried trying? Often that helps.”)